We summarize the main tools in Wu-Zhu [1]} to prove Theorem REF  in this section. It consists of four subsections, containing basic facts about calculus on graphs, properties of geodesic triangles and geodesic triangulations, and the curvature deformation with respect to discrete conformal factors. In addition, the last subsection provides a proof of a cubic estimation for controlling the deviation between smooth conformal deformation of metrics and discrete conformal deformation of metrics.
